How to calculate this type of integral

In summary, the integral in question is $\int\frac{{y}^{3}}{(196 - {y}^{2})\times \sqrt{196 - {y}^{2} - {a + y}^{2}}}$. A possible approach to solving it is to make the substitution $u = 196 - y^2$ and then use the relationship $\mathrm{d}u = -2\,y\,\mathrm{d}y$. However, further work is needed to properly solve the integral.
  • #1
zhaojx84
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Could anyone can tell me how to calculate this type of intergretion. Thanks very much
$$\int\frac{{y}^{3}}{(196 - {y}^{2})\times \sqrt{196 - {y}^{2} - {a + y}^{2}}}$$
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  • #2
Re: How to calculate this type of intergretion

Your integral does not match the integral in the picture. Which one are you trying to evaluate?
 
  • #3
zhaojx84 said:
Could anyone can tell me how to calculate this type of intergretion. Thanks very much
$$\int\frac{{y}^{3}}{(196 - {y}^{2})\times \sqrt{196 - {y}^{2} - {a + y}^{2}}}$$

I'll assume that the picture is the actual integral you are trying to solve...

$\displaystyle \begin{align*} \int{ \frac{y^3}{\left( 196 - y^2 \right) \,\sqrt{-\left( a + y \right) ^2 - y^2 + 196}} \,\mathrm{d}y } &= -\frac{1}{2} \int{ \frac{y^2}{\left( 196 - y^2 \right) \,\sqrt{ -\left( a + y \right) ^2 - y^2 + 196 }} \,\left( -2\,y \right) \,\mathrm{d}y } \end{align*}$

Let $\displaystyle \begin{align*} u = 196 - y^2 \implies \mathrm{d}u = -2\,y\,\mathrm{d}y \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\frac{1}{2} \int{\frac{y^2}{\left( 196 - y^2 \right) \,\sqrt{-\left( a + y \right) ^2 - y^2 + 196}}\,\left( -2\,y \right) \,\mathrm{d}y} &= -\frac{1}{2} \int{ \frac{196 - u}{u\,\sqrt{ \left( a + u - 196 \right) ^2 - u }} \,\mathrm{d}u } \end{align*}$

Does this seem a bit more manageable?
 
  • #4
Re: How to calculate this type of intergretion

Euge said:
Your integral does not match the integral in the picture. Which one are you trying to evaluate?
The integration in the figure is what I want to calculate.
Thanks very much for your help.
 
  • #5
Prove It said:
I'll assume that the picture is the actual integral you are trying to solve...

$\displaystyle \begin{align*} \int{ \frac{y^3}{\left( 196 - y^2 \right) \,\sqrt{-\left( a + y \right) ^2 - y^2 + 196}} \,\mathrm{d}y } &= -\frac{1}{2} \int{ \frac{y^2}{\left( 196 - y^2 \right) \,\sqrt{ -\left( a + y \right) ^2 - y^2 + 196 }} \,\left( -2\,y \right) \,\mathrm{d}y } \end{align*}$

Let $\displaystyle \begin{align*} u = 196 - y^2 \implies \mathrm{d}u = -2\,y\,\mathrm{d}y \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\frac{1}{2} \int{\frac{y^2}{\left( 196 - y^2 \right) \,\sqrt{-\left( a + y \right) ^2 - y^2 + 196}}\,\left( -2\,y \right) \,\mathrm{d}y} &= -\frac{1}{2} \int{ \frac{196 - u}{u\,\sqrt{ \left( a + u - 196 \right) ^2 - u }} \,\mathrm{d}u } \end{align*}$

Does this seem a bit more manageable?
Thanks for your help.
However, the substitution in the sqrt root is not correct.
 
  • #6
zhaojx84 said:
Thanks for your help.
However, the substitution in the sqrt root is not correct.

Yes I shouldn't try to help while tired haha. I'm sure you can fix it though :)
 
  • #7
Prove It said:
Yes I shouldn't try to help while tired haha. I'm sure you can fix it though :)
Could you help me later? I cannot figure it out. Thanks very much!
 

FAQ: How to calculate this type of integral

How do I determine which method to use when calculating an integral?

There are several methods for calculating integrals, including substitution, integration by parts, and trigonometric substitution. The best method to use will depend on the specific integral you are trying to solve. It is important to familiarize yourself with all of the methods and practice using them in order to determine the most efficient approach for a given integral.

Can I use a calculator to solve integrals?

While some basic integrals can be solved using a calculator, more complex integrals will require knowledge of integration techniques and cannot be accurately calculated using just a calculator. It is important to understand the concepts and techniques behind integration in order to successfully solve integrals.

How do I know if my integral is solvable?

Not all integrals can be solved using standard techniques. Some integrals may require more advanced techniques or may not have a closed form solution. It is always important to double check your work and make sure that your solution makes sense in the context of the problem.

Do I need to memorize integration formulas?

While it is helpful to have a basic understanding of common integration formulas, such as the power rule and trigonometric identities, it is not necessary to memorize them. As long as you understand the concepts behind integration, you will be able to apply the appropriate techniques to solve integrals.

Is there a shortcut for solving integrals?

There is no one-size-fits-all shortcut for solving integrals. Each integral will require a different approach and it is important to carefully consider the specific problem at hand. However, with practice and a good understanding of integration techniques, you will become more efficient at solving integrals.

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